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 A097781 Chebyshev polynomials S(n,27) with Diophantine property. 7
 1, 27, 728, 19629, 529255, 14270256, 384767657, 10374456483, 279725557384, 7542215592885, 203360095450511, 5483180361570912, 147842509666964113, 3986264580646460139, 107481301167787459640, 2898008866949614950141 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS All positive integer solutions of Pell equation b(n)^2 - 725*a(n)^2 = +4 together with b(n)=A090248(n+1), n>=0. Note that D=725=29*5^2 is not squarefree. For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 27's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011 For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,26}. - Milan Janjic, Jan 26 2015 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..700 R. Flórez, R. A. Higuita, A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014). Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (27,-1). FORMULA a(n) = S(n, 27) = U(n, 27/2) = S(2*n+1, sqrt(29))/sqrt(29) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x). a(n) = 27*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=27; a(-1)=0. a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (27+5*sqrt(29))/2 and am = (27-5*sqrt(29))/2. G.f.: 1/(1-27*x+x^2). a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*26^k. - Philippe Deléham, Feb 10 2012 Product {n >= 0} (1 + 1/a(n)) = 1/5*(5 + sqrt(29)). - Peter Bala, Dec 23 2012 Product {n >= 1} (1 - 1/a(n)) = 5/54*(5 + sqrt(29)). - Peter Bala, Dec 23 2012 EXAMPLE (x,y) = (27;1), (727;27), (19602;728), ... give the positive integer solutions to x^2 - 29*(5*y)^2 =+4. MAPLE with (combinat):seq(fibonacci(2*n, 5)/5, n=1..16); # Zerinvary Lajos, Apr 20 2008 MATHEMATICA Join[{a=1, b=27}, Table[c=27*b-a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *) CoefficientList[Series[1/(1 - 27 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 24 2012 *) PROG (Sage) [lucas_number1(n, 27, 1) for n in range(1, 20)] # Zerinvary Lajos, Jun 25 2008 (Magma) I:=[1, 27, 728]; [n le 3 select I[n] else 27*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012 CROSSREFS Cf. A078362, A078366. Sequence in context: A170746 A218729 A171332 * A223656 A073537 A016947 Adjacent sequences: A097778 A097779 A097780 * A097782 A097783 A097784 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Aug 31 2004 STATUS approved

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Last modified March 31 15:29 EDT 2023. Contains 361668 sequences. (Running on oeis4.)